| L(s) = 1 | + 2-s + 2.55·3-s + 4-s − 3.66·5-s + 2.55·6-s − 7-s + 8-s + 3.50·9-s − 3.66·10-s + 0.337·11-s + 2.55·12-s + 0.0496·13-s − 14-s − 9.36·15-s + 16-s + 2.81·17-s + 3.50·18-s − 0.175·19-s − 3.66·20-s − 2.55·21-s + 0.337·22-s + 2.49·23-s + 2.55·24-s + 8.46·25-s + 0.0496·26-s + 1.29·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.47·3-s + 0.5·4-s − 1.64·5-s + 1.04·6-s − 0.377·7-s + 0.353·8-s + 1.16·9-s − 1.16·10-s + 0.101·11-s + 0.736·12-s + 0.0137·13-s − 0.267·14-s − 2.41·15-s + 0.250·16-s + 0.683·17-s + 0.826·18-s − 0.0402·19-s − 0.820·20-s − 0.556·21-s + 0.0719·22-s + 0.520·23-s + 0.520·24-s + 1.69·25-s + 0.00973·26-s + 0.248·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.991019591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.991019591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 11 | \( 1 - 0.337T + 11T^{2} \) |
| 13 | \( 1 - 0.0496T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 + 0.175T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 - 0.466T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 4.77T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.29T + 73T^{2} \) |
| 79 | \( 1 - 2.19T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82058953861096539353432758574, −7.59475265727144381351812858483, −6.92004082542572179527514077926, −5.96317767498630920663343985276, −4.87835791381337146752933543797, −4.13718738323008120925333483905, −3.58719289352780281490447868013, −3.07727878190250432633215224749, −2.29390992556481554013695381427, −0.881231368387162953398886879256,
0.881231368387162953398886879256, 2.29390992556481554013695381427, 3.07727878190250432633215224749, 3.58719289352780281490447868013, 4.13718738323008120925333483905, 4.87835791381337146752933543797, 5.96317767498630920663343985276, 6.92004082542572179527514077926, 7.59475265727144381351812858483, 7.82058953861096539353432758574
